Higher Kaplansky theory
Date
2012-09-19
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Masters
Abstract
When studying the structure of a valued field $(K,v)$, immediate extensions are of special interest since they have the same value group and the same residue field as the ground field. One immediate extension that is of particular interest is the henselization $(K,v)^h$ of $(K,v)$ as it is a minimal immediate algebraic extension satisfying Hensel's Lemma, which in turn allows us to study the algebraic structure of a valued field through its residue field.
Kaplansky's work, based on earlier work of Ostrowski, laid the foundations for the understanding of immediate extensions. Here we present a continuation of Kaplansky's work, which allows us to determine special properties of elements in immediate extensions. As a tool to study there properties we introduce the notion of approximation types which represent an alternative, and in some sense an improvement, to the pseudo-convergent sequences used by Kaplansky. As a special interest to F.--V. Kuhlmann's work on henselian rationality over tame fields, we willinvestigate the question when an immediate valued function field of transcendence degree 1 is henselian rational (i.e., generated, modulo henselization, by one element). Henselian rationality is central in F.--V. Kuhlmann's work on local uniformization which is a local form of resolution of singularities.
Every immediate algebraic approximation type \bA over a valued field $(K,v)$, has a class of monic polynomials of minimal degree whose value is not fixed by \bA. Such polynomials are called associated minimal polynomials for \bA and Kaplansky in [4] stated a Theorem 10 indicating that easy normal forms can be determined for these polynomials. By generalizing Kaplansky's approach, we will show in Chapter 5 how such forms can be obtained.
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Keywords
local uniformization, henselian rationality, Kaplansky theory, approximation types
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Degree
Master of Science (M.Sc.)
Department
Mathematics and Statistics
Program
Mathematics